Sums of (x_i - a) or (x_i - a)^2 about given points: find the mean, variance or the constants
This type of question, asked 6 times: 2021 Paper 26, 29 Jul 2022, 1 Feb 2024, 29 Jan 2025, 4 Apr 2026 S2, 6 Apr 2026 S1. All 6 questions below, oldest first.
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Question 1
2021 Paper 26
83
Let X1,X2,………..X18 be eighteen observation such that ∑i=118(Xi−α)=36 and ∑i=118(Xi−β)2=90, where α and β are distinct real numbers. If the standard deviation of these observations is 1 , then the value of ∣α−β∣ is
Question 2
29 Jul 2022
82
Let the mean and the variance of 20 observations x1,x2,…x20 be 15 and 9 , respectively. For α∈R, if the mean of (x1+α)2,(x2+α)2,…,(x20+α)2 is 178 , then the square of the maximum value of α is equal to
Question 3
1 Feb 2024
168
Consider 10 observation x1,x2,…,x10 such that ∑i=110(xi−α)=2 and ∑i=110(xi−β)2=40, where α,β are positive integers. Let the mean and the variance of the observations be 56 and 2584 respectively. Then αβ is equal to:
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Question 4
29 Jan 2025
68
Let x1,x2,……x10 be ten observations such that ∑i=110(xi−2)=30,∑i=110(xi−β)2=98,β>2 and their variance is 54. If μ and σ2 are respectively the mean and the variance of 2(x1−1)+4β,2(x2−1)+4β,….,2(x10−1)+4β, then σ2βμ is equal to :
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Question 5
4 Apr 2026 S2
58
For 10 observations x1,x2,….,x10, if ∑i=110(xi+2)2=180 and ∑i=110(xi−1)2=90, then their standard deviation is :
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Question 6
6 Apr 2026 S1
59
A data consists of 20 observations x1,x2,…..,x20. If ∑i=120(xi+5)2=2500 and ∑i=120(xi−5)2=100, then the ratio of mean to standard deviation of this data is :
